Variable Hardy Spaces on Non-tangentially Accessible Domains and Their Applications to Div-Curl Lemma

Let n ≥ 2 , Ω ⊂ R n be a bounded non-tangentially accessible domain, and p ( · ) : R n → ( 0 , ∞ ) a variable exponent function satisfying 0 < p - ≤ p + < ∞ , where p - : = ess inf x ∈ R n p ( x ) and p + : = ess sup x ∈ R n p ( x ) . Assume that L D is a second order divergence form elliptic operator having real-valued, bounded, and measurable coefficients on L 2 ( Ω ) with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors establish the molecular characterization for the variable Hardy space H L D p ( · ) ( Ω ) associated with L D . In particular, when L D is non-negative and self-adjoint, the authors further obtain the atomic and the maximal function characterization of H L D p ( · ) ( Ω ) . Secondly, the authors introduce the “geometrical” variable Hardy space H r p ( · ) ( Ω ) and its local version h r p ( · ) ( Ω ) by restricting any element of the variable Hardy space H p ( · ) ( R n ) and its local version h p ( · ) ( R n ) , respectively, to Ω , and then show that, when p - ∈ ( n n + θ , 1 ] , H p ( · ) ( Ω ) = H r p ( · ) ( Ω ) = H L D p ( · ) ( Ω ) = h r p ( · ) ( Ω ) with equivalent quasi-norms, where H p ( · ) ( Ω ) denotes the variable Hardy space on Ω and θ ∈ ( 0 , 1 ] is the critical index of Hölder continuity for the heat kernels { p t , L D } t > 0 generated by L D . Thirdly, as applications, the authors prove the boundedness of the Riesz transform ∇ L D - 1 / 2 on the variable Lebesgue space L p ( · ) ( Ω ) when 1 < p - ≤ p + ≤ 2 , and from H L D p ( · ) ( Ω ) to L p ( · ) ( Ω ) when 0 < p - ≤ p + ≤ 1 , or to H p ( · ) ( Ω ) when n n + 1 < p - ≤ p + ≤ 1 . Meanwhile, the authors also show the boundedness of the fractional integral L D - β from L p ( · ) ( Ω ) to L q ( · ) ( Ω ) , when 1 < p ( · ) < q ( · ) < ∞ , and from H L D p ( · ) ( Ω ) to L q ( · ) ( Ω ) , when 0 < p ( · ) ≤ 1 < q ( · ) < ∞ , or to H L D q ( · ) ( Ω ) , when 0 < p ( · ) < q ( · ) ≤ 1 , where β ∈ ( 0 , n 2 ) and 1 p ( · ) - 1 q ( · ) = 2 β n . As a corollary, global gradient estimates, in both L p ( · ) ( Ω ) when 1 < p - ≤ p + ≤ 2 and H p ( · ) ( Ω ) when n n + 1 < p - ≤ p + ≤ 1 , for the inhomogeneous Dirichlet problem of L D on Ω are obtained. Finally, a div-curl lemma in the space H r p ( · ) ( Ω ) is established.